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A function is what the table says
Two circuits that give the same output for every input compute the same function, however differently they are wired. The table compares them without opening either.
Lesson · Mind, Language and Meaning · Mind
Functionalism and Multiple Realisability
Functionalism says a mental state is whatever plays a certain role, whatever it is made of. Write two circuits that compute the same function, check in the table that they agree in every row, and see what follows for pain being C-fibre firing.
same table, different circuit
(p ∧ q) ∨ (p ∧ r) three gates
p ∧ (q ∨ r) two gates
equal in all eight rows
functionalism: the state is the function
One function, two circuits
Core ideas
What this lesson establishes
A truth table is the smallest honest example of a function that can be built more than one way. The idea of the lesson is in that gap.
2
Realisability is the many wirings of one table
A state defined by its role can be realised by different parts: neurons in a person, silicon in a machine, valves in an imagined plumbing. That is multiple realisability, and the argument against identifying a state with one kind of part.
3
Agreeing on the inputs you tried is not agreeing
Two functions can agree on most rows and differ on one. A test that never reaches that row, whether a behavioural test or a table checked halfway, cannot tell them apart.
The material
The state as the function
Functionalism, then the circuit example, then the thing the example leaves out.
Definition · Functionalism
Functionalism says that a mental state is defined by the role it plays: what brings it about, what it brings about, and how it combines with other states. Anything that has states playing those roles is in the state, whatever it is made of.
If that is right, the identity theory that the physicalist reached for in “Dualism and the Conceivability Argument” was aiming at the wrong level. Pain is not the firing of C-fibres, because an octopus, which has no C-fibres, or a machine, which has no neurons at all, might be in pain by having something in the right role. One state, many realisations: that is multiple realisability.
A truth table shows the idea at its smallest scale. Here are two circuits with three inputs p, q and r. The first uses three gates, two of conjunction and one of disjunction: (p ∧ q) ∨ (p ∧ r). The second uses two, one of disjunction and one of conjunction: p ∧ (q ∨ r). They are wired differently. Put them in the lab and they agree in all eight rows, true in the same three and false in the same five.
So as functions they are one. Anything that treats an input according to this table is doing what both do, and nothing about the output reveals which circuit produced it. A functionalist says the same of minds: the state is the function, and the circuit is its realisation.
Other pairs in the lab show the variety. The circuit ¬(¬p ∨ ¬q) has a negation at each input, a disjunction and a negation at the output, and computes exactly p ∧ q, which is De Morgan's law as wiring. The circuit (p → q) ∧ (q → p) computes p ↔ q, again by a different route.
The same lab shows what a behavioural test can miss. The functions p ∧ q and p ∧ (q ∨ r) differ in one row only, where p and r are true and q false. Feed them every other input and they behave alike. A tester who never tries that input concludes they are the same, and is wrong.
What the example leaves out matters. A truth table is a function of the present inputs, and a mind has states that depend on history, so real functionalism describes machines with internal states, not tables. The lesson takes the smallest case in which the point can be checked row by row. And it does not show that the role is all there is to pain. That is what the next lessons press from two sides.
Interactive
Truth table
Every assignment, evaluated
Method
Checking that two circuits compute one function
Four steps, in this order. Only the last speaks to functionalism.
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Write each circuit as a formula
One formula per circuit, over the same input letters. A letter that one circuit never reads is still an input to the table.
2
Fill both tables over all inputs
Every assignment of true and false to every input letter, not a selection. A shortcut here is the mistake the lesson is about.
3
Compare the output columns row by row
If one row differs, the functions differ, and you have the input that tells them apart. If none does, they are one function.
4
Say what is shared and what is not
The shared thing is the table, and it is the candidate for the state. The unshared things, the number and kind of gates, are the realisations.
Same function, different circuit
Score 0 / 0Choose the answer you can justify.
Three gates against two
The two circuits compared in all eight rows of p, q, r.
A = (p ∧ q) ∨ (p ∧ r) three gates
B = p ∧ (q ∨ r) two gates
true rows of A: TTT TTF TFT
true rows of B: TTT TTF TFT
other five rows: A false, B false
differing rows: none
Different circuits, one function. The factoring of p out of the disjunction is the whole of the difference, and the table does not see it. A functionalist reads that as the point: what makes something the state is the table, not the factoring.
Common mistakes
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Thinking that the same behaviour means the same mechanismThe two circuits agree in all eight rows and are built from different gates, three against two. Output depends on the function, and a function can be reached by more than one route. That is why a functionalist does not identify a state with the parts that happen to realise it.
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Taking agreement on the inputs tried for agreementThe functions p ∧ q and p ∧ (q ∨ r) agree in seven rows and differ in the eighth. A behavioural test that skips the eighth input rates them equal. The judge of “Behaviourism and the Turing Test” is in that position: ten answers heard are ten rows of a table whose other rows were never tried.
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Reading functionalism as saying that the material never mattersThe claim is that the material does not decide which state it is. It can still decide whether anything plays the role at all: a circuit wired as p ∧ q does not realise the function p ∧ (q ∨ r), whatever it is made of.
Check your understanding
✓
Finish when you can show two realisations and say what they share.
Given a function, you should be able to write two different circuits for it, check by table that they agree in every row, and say what the claim that a state is a function amounts to when applied to the two.
Note: The circuits here have no memory. A mind, and a machine that functionalism is about, does, so the lesson is the idea at its smallest, not a theory of mind.